Simplify the following expression and state the condition under which the simplification is valid. $q = \dfrac{-10a^2 - 60a + 160}{-3a^2 - 15a + 72}$
Answer: First factor out the greatest common factors in the numerator and in the denominator. $ q = \dfrac {-10(a^2 + 6a - 16)} {-3(a^2 + 5a - 24)} $ $ q = \dfrac{10}{3} \cdot \dfrac{a^2 + 6a - 16}{a^2 + 5a - 24} $ Next factor the numerator and denominator. $ q = \dfrac{10}{3} \cdot \dfrac{(a + 8)(a - 2)}{(a + 8)(a - 3)}$ Assuming $a \neq -8$ , we can cancel the $a + 8$ $ q = \dfrac{10}{3} \cdot \dfrac{a - 2}{a - 3}$ Therefore: $ q = \dfrac{ 10(a - 2)}{ 3(a - 3)}$, $a \neq -8$